A382103 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.

3, 4, 7, 8, 5, 4, 8, 4, 5, 1, 3, 7, 4, 5, 3, 8, 5, 7, 3, 7, 3, 0, 6, 3, 9, 4, 9, 2, 2, 1, 9, 9, 9, 4, 0, 7, 2, 3, 5, 3, 4, 8, 6, 9, 5, 8, 3, 3, 8, 9, 3, 5, 4, 0, 4, 9, 2, 5, 2, 9, 3, 1, 9, 5, 1, 8, 7, 5, 1, 8, 6, 7, 4, 6, 5, 9, 1, 0, 3, 5, 1, 7, 2, 1, 9, 8, 3

Offset

0,1

Comments

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:

k | zeros | corresponding weights for Legendre-Gauss quadrature

---+---------------------------+----------------------------------------------------

2 | A020760 | A000007*10

3 | A010513/10 | A010716

4 | A372267, A372268 | this sequence, A382104

5 | A372269, A372270 | A382105, A382106

6 | A372271, A372272, A372273 | A382107, A382686, A382687

7 | A372274, A372275, A372276 | A382688, A382689, A382690

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=4.

A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.

Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Index entries for algebraic numbers, degree 2.

Formula

Equals 1/2 - (1/6)*sqrt(5/6).

Satisfies 216*x^2-216*x+49=0. - R. J. Mathar, Feb 27 2026

Example

0.34785484513745385737306394922199940723534869583389...

Mathematica

RealDigits[1/2 - Sqrt[5/6]/6, 10, 120][[1]] (* Amiram Eldar, Mar 24 2025 *)

Prog

(PARI) .5-sqrt(5/216) \\ Charles R Greathouse IV, Nov 17 2025

Crossrefs

Cf. A372267.

Sequence in context: A056007 A316973 A213622 * A246847 A193406 A104426

Adjacent sequences: A382100 A382101 A382102 * A382104 A382105 A382106

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 15 2025

Status

approved